Pseudoconvex regions of finite D'Angelo type in four dimensional almost complex manifolds
aa r X i v : . [ m a t h . C V ] O c t PSEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE IN FOUR DIMENSIONALALMOST COMPLEX MANIFOLDS
FLORIAN BERTRANDA
BSTRACT . Let D be a J -pseudoconvex region in a smooth almost complex manifold ( M, J ) of real dimen-sion four. We construct a local peak J -plurisubharmonic function at every point p ∈ bD of finite D’Angelotype. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashihyperbolicity of D at p . In case the point p is of D’Angelo type less than or equal to four, or the approach isnontangential, we provide sharp estimates of the Kobayashi pseudometric. C ONTENTS
Introduction
1. Preliminaries
2. Construction of a local peak plurisubharmonic function
3. Estimates of the Kobayashi pseudometric
4. Sharp estimates of the Kobayashi pseudometric
5. Appendix : Convergence of the structures involved by the scaling method. References
NTRODUCTION
Analysis on almost complex manifolds recently became a fondamental tool in symplectic geometry withthe work of M.Gromov in [15]. The local existence of pseudoholomorphic discs proved by A.Nijenhuis-W.Woolf in their paper [21], allows to define the Kobayashi pseudometric, which is crucial for local analysis.In the present paper we study the behaviour of the Kobayashi pseudometric of a J -pseudoconvex re-gion of finite D’Angelo type in an almost complex manifold ( M, J ) of dimension four. Finite D’Angelo Mathematics Subject Classification.
Primary 32Q60, 32T25, 32T40, 32Q45, 32Q65.
Key words and phrases.
Almost complex structure, peak plurisubharmonic functions, Kobayashi pseudometric, D’Angelo type. type appeared naturally in complex manifolds when considering the boundary behaviour of the ∂ operator(see [7],[8],[18],[4]). Moreover on complex manifolds of dimension two, the D’Angelo type unifies manytype conditions as the finite regular type. Finite regular type was recently characterized intrinsically by J.-F.Barrault-E.Mazzilli [1] by means of Lie brackets, which generalizes in the non integrable case, a result ofT.Bloom-I.Graham [4].Our main result is the construction of a local peak J -plurisubharmonic function on pseudoconvex regionsprovided by Theorem A (see also Theorem 2.6): Theorem A . Let D = { ρ < } be a domain of finite D’Angelo type in an almost complex manifold ( M, J ) of dimension four. We suppose that ρ is a C defining function of D , J -plurisubharmonic on a neighborhoodof D . Let p ∈ ∂D be a boundary point. Then there exists a local peak J -plurisubharmonic function at p . Theorem A allows to localize pseudoholomorphic discs and to obtain lower estimates of the Kobayashipseudometric which provide the local Kobayashi hyperbolicity of J -pseudoconvex regions of D’Angelotype m (Proposition 3.4 and Proposition 3.10). As an application we prove the / m -H ¨older extensionof biholomorphisms up to the boundary (Proposition 3.9). In order to obtain sharp lower estimates of theKobayashi pseudometric similar to those given in complex manifolds by D.Catlin [5] (see also [3]), weconsider a natural scaling method. However this reveals the fact that for a domain of finite D’Angelo typegreater than four, the sequence of almost complex structures obtained by any polynomial scaling processdoes not converge generically to the standard structure; this is presented in the Appendix. This may berelated to the fact that finite D’Angelo type is based on purely complex considerations, as the boundarybehaviour of the Cauchy-Riemann equations. Hence we provide sharp lower estimates of the Kobayashipseudometric for a region of finite D’Angelo type four (see also Theorem 4.1): Theorem B . Let D = { ρ < } be a relatively compact domain of finite D’Angelo type less than or equalto four in an almost complex manifold ( M, J ) of dimension four, where ρ is a C defining function of D , J -plurisubharmonic on a neighborhood of D . Then there is a positive constant C with the following property:for every p ∈ D and every v ∈ T p M there exists a diffeomophism Φ p ∗ in a neighborhood U of p , such that: K ( D,J ) ( p, v ) ≥ C | ( d p Φ p ∗ v ) || ρ ( p ) | + | ( d p Φ p ∗ v ) || ρ ( p ) | ! . We point out that the approach we use, based on some renormalization principle of pseudoholomorphicdiscs, gives also a different proof of precise lower estimates obtained by H.Gaussier-A.Sukhov in [12]for strictly J -pseudoconvex domains in arbitrary dimension. As an application of Theorem B, we obtainthe (local) complete hyperbolicity of J -pseudoconvex regions of D’Angelo type less than or equal to four(Corollary 4.5) and we give a Wong-Rosay theorem for regions with noncompact automorphisms group(Corollary 4.6).Finally, in order to obtain precise estimates near a point of arbitrary finite D’Angelo type, we are interestedin the nontangential behaviour of the Kobayashi pseudometric (see also Theorem 4.7): Theorem C . Let D = { ρ < } be a domain of finite D’Angelo type in an almost complex manifold ( M, J ) of dimension four, where ρ is a C defining function of D , J -plurisubharmonic on a neighborhood of D .Let q ∈ ∂D be a boundary point of D’Angelo type m and let Λ ⊂ D be a cone with vertex at q and axisthe inward normal axis. Then there exists a positive constant C such that for every p ∈ D ∩ Λ and every SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 3 v = v n + v t ∈ T p M : K ( D,J ) ( p, v ) ≥ C | v n || ρ ( p ) | m + | v t || ρ ( p ) | ! , where v n and v t are the normal and the tangential parts of v with respect to q .
1. P
RELIMINARIES
We denote by ∆ the unit disc of C and by ∆ r the disc of C centered at the origin of radius r > .1.1. Almost complex manifolds and pseudoholomorphic discs. An almost complex structure J on a realsmooth manifold M is a (1 , tensor field which satisfies J = − Id . We suppose that J is smooth. Thepair ( M, J ) is called an almost complex manifold . We denote by J st the standard integrable structure on C n for every n . A differentiable map f : ( M ′ , J ′ ) −→ ( M, J ) beetwen two almost complex manifolds is saidto be ( J ′ , J ) -holomorphic if J ( f ( p )) ◦ d p f = d p f ◦ J ′ ( p ) , for every p ∈ M ′ . In case M ′ = ∆ ⊂ C , sucha map is called a pseudoholomorphic disc . If f : ( M, J ) −→ M ′ is a diffeomorphism, we define an almostcomplex structure, f ∗ J , on M ′ as the direct image of J by f : f ∗ J ( q ) := d f − ( q ) f ◦ J (cid:0) f − ( q ) (cid:1) ◦ d q f − , for every q ∈ M ′ .The following lemma (see [12]) states that locally any almost complex manifold can be seen as the unitball of C n endowed with a small smooth pertubation of the standard integrable structure J st . Lemma 1.1.
Let ( M, J ) be an almost complex manifold, with J of class C k , k ≥ . Then for everypoint p ∈ M and every λ > there exist a neighborhood U of p and a coordinate diffeomorphism z : U → B centered a p (ie z ( p ) = 0 ) such that the direct image of J satisfies z ∗ J (0) = J st and || z ∗ ( J ) − J st || C k ( ¯ B ) ≤ λ . This is simply done by considering a local chart z : U → B centered a p (ie z ( p ) = 0 ), composing it witha linear diffeomorphism to insure z ∗ J (0) = J st and dilating coordinates.So let J be an almost complex structure defined in a neighborhood U of the origin in R n , and such that J is sufficiently closed to the standard structure in uniform norm on the closure U of U . The J -holomorphyequation for a pseudoholomorphic disc u : ∆ → U ⊆ R n is given by(1.1) ∂u∂y − J ( u ) ∂u∂x = 0 . According to [21], for every p ∈ M , there is a neighborhood V of zero in T p M , such that for every v ∈ V , there is a J -holomorphic disc u satisfying u (0) = p and d u ( ∂/∂x ) = v .1.2. Levi geometry.
Let ρ be a C real valued function on a smooth almost complex manifold ( M, J ) . Wedenote by d cJ ρ the differential form defined by d cJ ρ ( v ) := − dρ ( J v ) , where v is a section of T M . The
Levi form of ρ at a point p ∈ M and a vector v ∈ T p M is defined by L J ρ ( p, v ) := d ( d cJ ρ ) ( p ) ( v, J ( p ) v ) = dd cJ ρ ( p ) ( v, J ( p ) v ) . In case ( M, J ) = ( C n , J st ) , then L J st ρ is, up to a positive multiplicative constant, the usual standard Leviform : L J st ρ ( p, v ) = 4 X ∂ ρ∂z j ∂z k v j v k . FLORIAN BERTRAND
We investigate now how close is the Levi form with respect to J from the standard Levi form. For p ∈ M and v ∈ T p M , we easily get :(1.2) L J ρ ( p, v ) = L J st ρ ( p, v ) + d ( d cJ − d cJ st ) ρ ( p )( v, J ( p ) v ) + dd cJ st ρ ( p )( v, J ( p ) − J st ) v ) . In local coordinates ( t , t , · · · , t n ) of R n , (1.2) may be written as follows L J ρ ( p, v ) = L J st ρ ( p, v ) + t v ( A − t A ) J ( p ) v + t ( J ( p ) − J st ) vDJ st v + t ( J ( p ) − J st ) vD ( J ( p ) − J st ) v (1.3)where A := X i ∂u∂t i ∂J i,j ∂t k ! ≤ j,k ≤ n and D := (cid:18) ∂ u∂t j ∂t k (cid:19) ≤ j,k ≤ n . Let f be a ( J ′ , J ) -biholomorphism from ( M ′ , J ′ ) to ( M, J ) . Then for every p ∈ M and every v ∈ T p M : L J ′ ρ ( p, v ) = L J ρ ◦ f − ( f ( p ) , d p f ( v )) . This expresses the invariance of the Levi form under pseudobiholomorphisms.The next proposition is useful in order to compute the Levi form (see [10], [16] and [17]).
Proposition 1.2.
Let p ∈ M and v ∈ T p M . Then L J ρ ( p, v ) = ∆ ( ρ ◦ u ) (0) , where u : ∆ → ( M, J ) is any J -holomorphic disc satisfying u (0) = p and d u ( ∂/∂ x ) = v . Proposition 1.2 leads to the following proposition-definition :
Proposition 1.3.
The two statements are equivalent : (1) ρ ◦ u is subharmonic for any J -holomorphic disc u : ∆ → M . (2) L J ρ ( p, v ) ≥ for every p ∈ M and every v ∈ T p M . If one of the previous statements is satisfied we say that ρ is J -plurisubharmonic . We say that ρ is strictly J -plurisubharmonic if L J ρ ( p, v ) is positive for any p ∈ M and any v ∈ T p M \ { } . J -plurisubharmonicfunctions play a very important role in almost complex geometry : they give attraction and localizationproperties for pseudoholomorphic discs. For this reason the construction of J -plurisubharmonic functionsis crucial.Similarly to the integrable case, one may define the notion of pseudoconvexity in almost complex mani-folds. Let D be a domain in ( M, J ) . We denote by T J ∂D := T ∂D ∩ J T ∂D the J -invariant subbundle of T ∂D.
Definition 1.4. (1) The domain D is J -pseudoconvex (resp. it strictly J -pseudoconvex) if L J ρ ( p, v ) ≥ (resp. > )for any p ∈ ∂D and v ∈ T Jp ∂D (resp. v ∈ T Jp ∂D \ { } ).(2) A J -pseudoconvex region is a domain D = { ρ < } where ρ is a C defining function, J -plurisubharmonic on a neighborhood of D .We recall that a defining function for D satisfies dρ = 0 on ∂D .The following Lemma is useful in order to compute the Levi form of some functions. SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 5
Lemma 1.5.
Assume that J is a diagonal almost complex structure on R that coincides with the standardstructure J on C × { } . To fix notations we suppose that its matricial representation is given by : J = a b c − a a b c − a . Then the Levi form of some smooth real valued function f at a point z = ( z , z ) and v = (1 , , , isequal to L J f ( z, v ) = − c ∆ f + O ( | z | )) . where ∆ f := ∂ f∂x ∂x + ∂ f∂y ∂y .Proof. Let us compute the Levi form of some smooth real valued function f at a point z = ( z , z ) and v = (1 , , , : c − L J f ( z, v ) = − ∆ f + (cid:20) − ∂ f∂x ∂y a + ∂ f∂x ∂x (1 + b ) + ∂ f∂y ∂y ( c − (cid:21) + ∂f∂x (cid:20) ∂b ∂x − ∂a ∂y (cid:21) + ∂f∂y (cid:20) ∂a ∂x + ∂c ∂y (cid:21) = − ∆ f + (cid:20) − ∂ f∂x ∂y O ( | z | ) + ∂ f∂x ∂x O ( | z | ) + ∂ f∂y ∂y O ( | z | ) (cid:21) + ∂f∂x O ( | z | ) + ∂f∂y O ( | z | )= − ∆ f + O ( | z | ) . (cid:3)
2. C
ONSTRUCTION OF A LOCAL PEAK PLURISUBHARMONIC FUNCTION
This section is devoted to the proof of Theorem A (see Theorem 2.6).2.1.
Pseudoconvex regions of finite D’Angelo type.
In this subsection we describe a pseudonconvex re-gion on a neighborhood of a boundary point of finite D’Angelo type. We point out that all our considerationsare purely local. Assume that D = { ρ < } is a J -pseudoconvex region in C and that the structure J isdefined on a fixed neighborhood U of D . We suppose that the origin is a boundary point of D . Definition 2.1.
Let u : (∆ , → (cid:0) R , , J (cid:1) be a J -holomorphic disc satisfying u (0) = 0 . The order ofcontact δ ( ∂D, u ) with ∂D at the origin is the degree of the first term in the Taylor expansion of ρ ◦ u . Wedenote by δ ( u ) the multiplicity of u at the origin.We now define the D’Angelo type and the regular type of the real hypersurface ∂D at the origin. Definition 2.2. (1) The D’Angelo type of ∂D at the origin is defined by: ∆ ( ∂D,
0) := sup (cid:26) δ ( ∂D, u ) δ ( u ) , u : ∆ → (cid:0) R , J (cid:1) J -holomorphic , u (0) = 0 (cid:27) . The point is a point of finite D’Angelo type m if ∆ ( ∂D,
0) = 2 m < + ∞ . FLORIAN BERTRAND (2) The regular type of ∂D at origin is defined by: ∆ ( ∂D,
0) := sup { δ ( ∂D, u ) , u : ∆ → (cid:0) R , J (cid:1) J -holomorphic ,u (0) = 0 , d u = 0 } . Since the regular type of ∂D at the origin consists in considering only regular discs we have:(2.1) ∆ ( ∂D, ≤ ∆ ( ∂D, . The type condition as defined in part 1 of Definition 2.2 was introduced by J.-P.D’Angelo [7], [8] whoproved that this coincides with the regular type in complex manifolds of dimension two. After Proposition2.3, we will also prove that the D’Angelo type and the regular type coincide in four dimensional almostcomplex manifolds (see Proposition 2.4).We suppose that the origin is a point of finite regular type. Then let u : ∆ → R be a regular J -holomorphic disc of maximal contact order m . We choose coordinates such that u is given by u ( ζ ) =( ζ, , J ( z ,
0) = J st and such that the complex tangent space T ∂D ∩ J (0) T ∂D is equal to { z = 0 } .Then by considering the family of vectors (1 , at base points (0 , t ) for t = 0 small enough, we obtain afamily of J holomorphic discs u t such that u t (0) = (0 , t ) and d u t ( ∂/∂ x ) = (0 , . Due to the parametersdependance of the solution to the J -holomorphy equation (1.1), we straighten these discs into the complexlines { z = t } . We then consider a transversal foliation by J -holomorphic discs and straighten these linesinto { z = c } . In these new coordinates still denoted by z , the matricial representation of J is diagonal:(2.2) J = a b c − a a b c − a . Since J ( z ,
0) = J st we have(2.3) J = J st + O ( | z | ) . In the next fundamental proposition we describe precisely the local expression of the defining function ρ . Proposition 2.3.
The J -plurisubharmonic defining function for the domain D has the following local ex-pression: ρ = ℜ ez + H m ( z , z ) + e H ( z , z ) + O (cid:0) | z | m +1 + | z || z | m + | z | (cid:1) where H m is a homogeneous polynomial of degree m , subharmonic which is not harmonic and e H ( z , z ) = ℜ e m − X k =1 ρ k z k z . Proof.
Since T ∂D ∩ J (0) T ∂D = { z = 0 } , we have ρ = ℜ ez + O ( k z k ) . Moreover the disc ζ ( ζ, being a regular J -holomorphic disc of maximal contact order m , the definingfunction ρ has the following local expression: ρ = ℜ ez + H m ( z , z ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) , where H m is a homogeneous polynomial of degree m . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 7
We prove that the polynomial H m is subharmonic using a standard dilation argument. Consider thenon-isotropic dilation of C Λ δ ( z , z ) := (cid:16) δ − m z , δ − z (cid:17) . Due to Proposition 1.2, the domain Λ δ ( D ) = { δ − (cid:0) ρ ◦ Λ − δ ( z , z ) (cid:1) < } is (Λ δ ) ∗ ( J ) -pseudoconvex. Moreover Λ δ ( D ) converges in the sense of local Hausdorff set convergence to ˜ D := { Re ( z ) + H m ( z , z ) < } , as δ tends to zero and the sequence of structures (Λ δ ) ∗ J converges to the standard structure J st . It followsthat the limit domain ˜ D is J st -pseudoconvex implying that H m is subharmonic.Now we prove H m that contains a nonharmonic part. By contradiction, we assume that H m is harmonic.Then H m can be written ℜ ez m . According to Proposition 1.1 of [17], and since the structure J is smooththere exists, for a sufficiently small λ > , a pseudoholomorphic disc u : ∆ → ( R , J ) such that: u (0) = 0 ∂u∂x (0) = (cid:16) λ m , , , (cid:17) ∂ k u∂x k (0) = (0 , , , , for < k < m∂ m u∂x m (0) = (0 , , − λ (2 m )! , . We prove that the contact order of such a regular disc u is greater than m which contradicts the fact that D is of regular type m . We denote by [ ρ ◦ u ] m the homogeneous part of degree m in the Taylor expansionof ρ ◦ u at the origin: [ ρ ◦ u ] m ( x, y ) = m X k =0 a k x k y m − k . Let us prove that a k = ∂ k ∂x k ∂ m − k ∂y m − k ρ ◦ u (0) is equal to zero for each ≤ k ≤ m .For a m , we have: ∂ m ∂x m ρ ◦ u (0) = ℜ e ∂ m ∂x m u (0) + ℜ e ∂ m ∂x m u m (0)= − λ (2 m )! + ℜ e ∂ m ∂x m u m (0) . Since u (0) = 0 , it follows that the only non vanishing term in ℜ e ∂ m ∂x m u m (0) is (2 m )! ℜ e (cid:18) ∂u ∂x (0) (cid:19) m = λ (2 m )! . This proves that a m = 0 . FLORIAN BERTRAND
Then let ≤ k < m : ∂ k ∂x k ∂ m − k ∂y m − k ρ ◦ u (0) = ℜ e ∂ k ∂x k ∂ m − k ∂y m − k u (0) + ℜ e ∂ k ∂x k ∂ m − k ∂y m − k u m (0) . For the same reason as previously, the only term to consider in ℜ e ∂ k ∂x k ∂ m − k ∂y m − k u m (0) is (2 m )! ℜ e (cid:18) ∂∂x u (0) (cid:19) k (cid:18) ∂∂y u (0) (cid:19) m − k = λ k m (2 m )! ℜ e (cid:18) ∂u ∂y (0) (cid:19) m − k . Then, since u is J -holomorphic, it satisfies the diagonal J -holomorphy equation: ∂u l ∂y = J l ( u ) ∂u l ∂x , for l = 1 , , where J l = (cid:18) a l b l c l − a l (cid:19) (see (2.2) for notations).It follows that λ k m (2 m )! ℜ e (cid:18) ∂u ∂y (0) (cid:19) m − k = λ k m (2 m )! ℜ e (cid:18) J ( u (0)) ∂u ∂x (0) (cid:19) m − k = λ (2 m )! ℜ e ( i ) m − k . Moreover due to the condition ∂ k u ∂x k (0) = (0 , , for ≤ k < m , it follows that the only part we needto consider in ∂ m − k ∂y m − k u (0) is J ( u ) ∂∂x ∂ m − k − ∂y m − k − u (0) and by induction ( J ( u )) m − k ∂ m − k ∂x m − k u (0) .Finally ℜ e ∂ k ∂x k ∂ m − k ∂y m − k u (0) = ℜ e ( J ( u (0))) m − k ∂ m u ∂x m (0)= − λ (2 m )! ℜ e (cid:16) J ( u (0)) m − k (1 , (cid:17) = − λ (2 m )! ℜ e ( i ) m − k . This proves that the homogeneous part [ ρ ◦ u ] m is equal to zero.For smaller order terms it is a direct consequence of u (0) = 0 and ∂ k u∂x k (0) = (0 , , , , for < k < m .It remains to prove there are no term ℜ eρ k z k z with k < m in the defining function ρ . This is done bycontradiction and by computing the Levi form of ρ at a point z = ( z , and at a vector v = ( X , , X , .Assume that ρ = ℜ ez + H m ( z , z ) + e H ( z , z ) + ℜ eρ k z k z + O (cid:16) | z | m +1 + | z || z | k +1 + | z | (cid:17) , with k < m . Replacing z by ( ρ k ) k z if necessary, we suppose ρ k = 1 . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 9
The Levi form of ℜ ez at a point z = ( z , and at a vector v = ( X , , X , is equal to L J ℜ ez ( z , v ) = (cid:20) ( a − a ) ( z ) ∂a ∂x ( z ) + c ( z ) ∂a ∂y ( z ) − c ( z ) ∂b ∂x ( z ) (cid:21) X X + c ( z ) (cid:20) ∂a ∂y ( z ) − ∂b ∂x ( z ) (cid:21) X . Due to (2.3) we have a ( z ) = a ( z ) = 0 ,c ( z ) = 1 ,∂a ∂y ( z ) = ∂b ∂x ( z ) = 0 . So the Levi form of ℜ ez at z = ( x , , , and at a vector v = ( X , , X , is L J ℜ ez ( z , v ) = (cid:20) ∂a ∂y ( z ) − ∂b ∂x ( z ) (cid:21) X . According to Lemma 1.5, the Levi form of H m + O ( | z | m +1 ) at z and v = ( X , , X , is equal to L J ( H m + O ( | z | m +1 )) ( z , v ) = ∆ (cid:0) H m + O ( | z | m +1 ) (cid:1) X + O ( | z | m − ) X X . According to the fact that the Levi form for the standard structure of e H ( z , z ) is identically equal tozero, and due to (1.3) and to (2.3), it follows that the Levi form of e H ( z , z ) at z is equal to L J e H ( z , v ) = O ( | z | ) X . Now the Levi form of O ( | z | ) is equal to L J O ( | z | ) ( z , v ) = O (1) X . And the Levi form of ℜ ez k z is equal L J ℜ ez k z ( z , v ) = ( k ℜ ez k − ) X X + O ( | z | k ) X . Finally the Levi form of the defining function ρ at a point z = ( z , and at a vector v = ( X , , X , is equal to: L J ρ ( z , v ) = O (cid:0) | z | m − (cid:1) X + h k ℜ ez k − + O ( | z | m − ) i X X + (cid:20) ∂a ∂y ( z ) − ∂b ∂x (0) + O (1) + O ( | z | ) (cid:21) X . It follows that since k < m there are z , X and X such that L J ρ ( z , v ) is negative, providing a contra-diction. (cid:3) Now we prove that the D’Angelo type coincides with the regular type in the non integrable case.
Proposition 2.4.
We have ∆ ( ∂D,
0) = ∆ ( ∂D, . Proof.
We suppose that the origin is a point of finite D’Angelo type. According to (2.1) we may write: ∆ ( ∂D,
0) = 2 m < + ∞ . So we may assume that u ( ζ ) = ( ζ, is a regular J -holomorphic disc of maximal contact order m , and thatthe structure J satisfies (2.2) and (2.3). Moreover the defining function ρ has the following local expression: ρ = ℜ ez + H m ( z , z ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) . Now consider a J -holomorphic disc v = ( f , g , f , g ) : (∆ , → (cid:0) R , , J (cid:1) of finite contact ordersatisfying v (0) = 0 and such that δ ( v ) ≥ (see definition 2.1 for notations).We set v := f + ig and v := f + ig . The J -holomorphy equation for the disc v is given by: a k ( v ) ∂f k ∂x + b k ( v ) ∂g k ∂x = ∂f k ∂y ,c k ( v ) ∂f k ∂x − a k ( v ) ∂g k ∂x = ∂g k ∂y , for k = 1 , . Since J ( v ) = J st + O ( | v | ) and δ ( v ) ≥ , it follows that:(2.4) δ ( v ) = δ ( f ) = δ ( g ) ,δ ( v ) = δ ( f ) = δ ( g ) . Then consider(2.5) ρ ◦ v ( ζ ) = f ( ζ ) + H m (cid:16) v ( ζ ) , v ( ζ ) (cid:17) + O (cid:0) | v ( ζ ) | m +1 + | v ( ζ ) |k v ( ζ ) k (cid:1) . Equation (2.4) implies that the term O ( | v |k v k ) in (2.5) vanishes to order larger than f . Case 1: δ ( f ) > δ ( H m ( v , v )) . In that case δ ( ∂D, u ) = δ ( H m ( v , v )) = 2 mδ ( v ) . Thus we get: δ ( ∂D, v ) δ ( v ) = 2 mδ ( v ) δ ( v ) = 2 m. Case 2: δ ( f ) ≤ δ ( H m ( v , v )) . We have two subcases. Subcase 2.1: f + H m ( v , v ) . Thus δ ( ∂D, u ) = δ ( ℜ ev ) = δ ( v ) , and so δ ( ∂D, v ) δ ( v ) = δ ( v ) δ ( v ) ≤ δ ( H m ( v , v )) δ ( v ) = 2 mδ ( v ) δ ( v ) . This means that: δ ( ∂D, v ) δ ( v ) = 1 if δ ( v ) = δ ( v ) or δ ( ∂D, v ) δ ( v ) ≤ m if δ ( v ) = δ ( v ) . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 11
Subcase 2.2: f + H m ( v , v ) ≡ . Let w : ∆ → (cid:0) R , J st (cid:1) be a standard holomorphic disc satisfying w (0) = 0 and: ∂ k w∂x k (0) = ∂ k v∂x k (0) , for k = 1 , · · · , mδ ( v ) . Since δ ( v ) = 2 mδ ( v ) = 2 mδ ( v ) < + ∞ and since J ( v ) = J st + O ( | v | ) ,any differentiation of J ( v ) , of order smaller than mδ ( v ) , is equal to zero. Combining this with the J -holomorphy equation (1.1) of v we obtain: ∂ k + l w∂x k ∂y l (0) = ∂ k + l v∂x k ∂y l (0) , for k + l = 1 , · · · , mδ ( v ) . Since ρ ◦ v vanishes to an order greater than mδ ( v ) at 0 and since it involvesonly the mδ ( v ) -jet of v , it follows that ρ ◦ w vanishes to an order greater than mδ ( v ) at 0. Finally wehave constructed a standard holomorphic disc w such that δ ( w ) = δ ( v ) ,δ ( ∂D, w ) > mδ ( w ) , which is not possible since, according Proposition 2.3, the type for the standard structure of ∂D at the originis equal to m . (cid:3) Construction of a local peak plurisubharmonic function.
We first give the definition of a local peak J -plurisubharmonic function for a domain D . Definition 2.5.
Let D be a domain in an almost complex manifold ( M, J ) . A function ϕ is called a localpeak J -plurisubharmonic function at a boundary point p ∈ ∂D if there exists a neighborhood U of p suchthat ϕ is continuous up to D ∩ U and satisfies:(1) ϕ is J -plurisubharmonic on D ∩ U ,(2) ϕ ( p ) = 0 ,(3) ϕ < on D ∩ U \{ p } . The existence of local peak J st -plurisubharmonic functions was first proved byE.Fornaess and N.Sibony in [11]. For almost complex manifolds the existence was proved by S.Ivashkovichand J.-P.Rosay in [17] whenever the domain is strictly J -pseudoconvex. In the next Proposition we state theexistence for J -pseudoconvex regions of finite D’Angelo type. As mentionned earlier our the considerationsare purely local. In particular, the assumptions of J -plurisubharmonicity and of finite D’Angelo type maybe restricted to a neighborhood of a boundary point. For convenience of writing, we state them globally. Theorem 2.6.
Let D = { ρ < } be a domain of finite D’Angelo type in a four dimensional almost complexmanifold ( M, J ) . We suppose that ρ is a C defining function of D , J -plurisubharmonic on a neighborhoodof D . Let p ∈ ∂D be a boundary point. Then there exists a local peak J -plurisubharmonic function at p .Proof. Since the existence of a local peak function near a boundary point of type was proved in [17], weassume that p is a boundary point of D’Angelo type m > . The problem being purely local we assumethat D ⊂ C and that p = 0 . According to Proposition 2.3 the defining function ρ has the following localexpression on a neighborhood U of the origin: ρ = ℜ ez + H m ( z , z ) + e H ( z , z ) + O (cid:0) | z | m +1 + | z || z | m + | z | (cid:1) where H m is a subharmonic polynomial containing a nonharmonic part, denoted by H ∗ m , and e H ( z , z ) = ℜ e m − X k =1 ρ k z k z . According to [11] (see Lemma 2.4), the polynomial H m satisfies the following Lemma: Lemma 2.7.
There exist a positive δ > and a smooth function g : R → R with period π with thefollowing properties: (1) − < g ( θ ) < − , (2) k g k < /δ , (3) max (cid:0) ∆ H m , ∆ (cid:0) k H ∗ m k g ( θ ) | z | m (cid:1)(cid:1) > δ k H ∗ m k| z | m − , for z = | z | e iθ = 0 and, (4) ∆ (cid:0) H m + δ k H ∗ m k g ( θ ) | z | m (cid:1) > δ k H ∗ m k| z | m − . We denote by P the function defined by P ( z , z ) := H m ( z , z ) + δ k H ∗ m k g ( θ ) | z | m . Theorem 2.6 will be proved by establishing the following claim.
Claim.
There are positive constants L and C such that the function ϕ := ℜ ez + 2 L ( ℜ ez ) − L ( ℑ mz ) + P ( z , z ) + e H ( z , z ) + C | z | | z | is a local peak J -plurisubharmonic function at the origin. Proof of the claim.
We first prove that the function ϕ is J -plurisubharmonic. We set: dd cJ ϕ = α dx ∧ dy + α dx ∧ dy + α dx ∧ dx + α dx ∧ dy + α dy ∧ dx + α dy ∧ dy , where α k , for k = 1 , · · · , , are real valued function. According to the matricial representation of J (see(2.2)), the Levi form of ϕ at a point z ∈ D ∩ U and at a vector v = ( X , Y , X , Y ) ∈ T z R can be written L J ϕ ( z, v ) = c α X − a α X Y − b α Y + β X X + β X Y + β Y X + β Y Y + c α X − a α X Y − b α Y , with β := α ( a − a ) + α c − α c β := − α ( a + a ) + α b − α c β := α ( a + a ) − α b + α c β := α ( a − a ) − α b + α b . Moreover due to (2.3) we have for k = 1 , a k = O ( | z | ) b k = − O ( | z | ) c k = 1 + O ( | z | ) . This implies that for k = 1 , : c k α k X k − a k α k X k Y k − b k α k Y k ≥ α k (cid:0) X k + Y k (cid:1) . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 13
Thus we obtain L J ϕ ( z, v ) ≥ α X + β X X + α X + α Y + β Y X + α X + α X + β X Y + α Y + α Y + β Y Y + α Y . In order to prove that ϕ is J -plurisubharmonic, we need to see that:(1) α k ≥ , for k = 1 , ,(2) β j ≤ α α , for j = 3 , · · · , .The coefficient α is obtained by the differentiation of ℜ ez , L ( ℜ ez ) − L ( ℑ mz ) , e H ( z , z ) and C | z | | z | . Hence we have for z sufficiently close to the origin α ≥ L > . The coefficient α is obtained by differentiating P , e H ( z , z ) and C | z | | z | . This is equal to α = ∆ P + O ( | z | m − | z | ) + O ( | z | ) + C | z | + O ( | z | ) ≥ δ k H ∗ m k | z | m − + C | z | , for z sufficiently small and C > large enough. Hence α is nonnegative.Finally it sufficient to prove that β j ≤ L (cid:18) δ k H ∗ m k | z | m − + C | z | (cid:19) , to insure the J -plurisubharmonicity of ϕ . The coefficient | β j | is equal to | β j | = O ( | z | ) + LO ( | z | ) + O ( | z | m − ) + CO ( | z || z | ) ≤ C ′ ( | z | + | z | m − ) , for a positive constant C ′ (not depending on L and C ). It follows that ϕ is J -plurisubharmonic on a neigh-borhood of the origin.We prove now that ϕ is local peak at the origin, that is there exists r > such that D ∩ { < k z k ≤ r } ⊂{ ϕ < } . Assuming that z ∈ { ρ = 0 } ∩ { < k z k ≤ r } we have: ϕ ( z ) = δ k H ∗ m k g ( θ ) | z | m + 2 L ( ℜ ez ) − L ( ℑ mz ) + C | z | | z | + O (cid:0) | z | m +1 (cid:1) + O ( | z || z | m ) + O (cid:0) | z | (cid:1) . Since g < − and increasing L if necessary we have O ( |ℑ mz || z | m ) ≤ − δ k H ∗ m k g ( θ ) | z | m + 12 L ( ℑ mz ) , whenever z is sufficiently close to the origin. Thus ϕ ( z ) ≤ − δ k H ∗ m k| z | m + (2 L + C | z | ) ( ℜ ez ) − L ( ℑ mz ) + C | z | ( ℑ mz ) + O (cid:0) | z | m +1 (cid:1) + O ( |ℜ ez || z | m ) + O (cid:0) | z | (cid:1) ≤ − δ k H ∗ m k| z | m + (2 L + C | z | ) ( ℜ ez ) − L ( ℑ mz ) + O ( |ℜ ez || z | m ) + O (cid:0) | z | (cid:1) . There is a positive constant C ′′ such that O (cid:0) | z | (cid:1) ≤ C ′′ |ℜ ez | + C ′′ |ℑ mz | . Thus increasing L if necessary: ϕ ( z ) ≤ − δ k H ∗ m k| z | m + (2 L + C | z | ) ( ℜ ez ) + O ( |ℜ ez | ) − (cid:18) L − C ′′ (cid:19) ( ℑ mz ) + O ( |ℜ ez || z | m ) + O ( |ℑ mz | k z k ) . ≤ − δ k H ∗ m k| z | m + (2 L + C | z | ) ( ℜ ez ) + O ( |ℜ ez | ) + O ( |ℜ ez || z | m ) − (cid:18) L − C ′′ (cid:19) ( ℑ mz ) . Since −ℜ ez (1 + O ( | z | )) = H m ( z , z ) + O (cid:0) | z | m +1 + |ℑ mz || z | + |ℑ mz | (cid:1) , we have ( ℜ ez ) (1 + O ( | z | )) = O (cid:0) | z | m + |ℑ mz || z | m +1 + |ℑ mz | k z k (cid:1) . We finally obtain for z small enough ϕ ( z ) ≤ − δ k H ∗ m k| z | m − (cid:18) L − C ′′ (cid:19) ( ℑ mz ) . Thus ϕ is negative for z ∈ { ρ = 0 } ∩ { < k z k ≤ r } , with r small enough. It follows that, reducing r ifnecessary, D ∩ { < k z k ≤ r } ⊂ { ϕ < } , which achieves the proof of the claim and of Theorem 2.6. (cid:3) We notice that in case L J ℜ ez ≡ , we may give a simpler expression for a local peak J -plurisubharmonic function. Proposition 2.8. If L J ℜ ez ≡ , then there exists a real positive number L such that the function ϕ := ℜ ez + 2 L ( ℜ ez ) − L ( z ) + P ( z , z ) is local peak J -plurisubharmonic at the origin. SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 15
3. E
STIMATES OF THE K OBAYASHI PSEUDOMETRIC
In this section we prove standard estimates of the Kobayashi pseudometric on J -pseudoconvex regionsof finite D’Angelo type in an almost complex manifold.3.1. The Kobayashi pseudometric.
The existence of local pseudoholomorphic discs proved in [21] allowsto define the
Kobayashi pseudometric K ( M,J ) for p ∈ M and v ∈ T p M : K ( M,J ) ( p, v ) := inf (cid:26) r > , u : ∆ → ( M, J ) J -holomorphic , u (0) = p, d u ( ∂/∂x ) = rv (cid:27) . Since the composition of pseudoholomorphic maps is still pseudoholomorphic, the Kobayashi (infinites-imal) pseudometric satisfies the following decreasing property :
Proposition 3.1.
Let f : ( M ′ , J ′ ) → ( M, J ) be a ( J ′ , J ) -holomorphic map. Then for any p ∈ M ′ and v ∈ T p M ′ we have K ( M ′ ,J ′ ) ( p, v ) ≥ K ( M,J ) ( f ( p ) , d p f ( v )) . Let d ( M,J ) be the integrated pseudodistance of K ( M,J ) : d ( M,J ) ( p, q ) := inf (cid:26)Z K ( M,J ) ( γ ( t ) , ˙ γ ( t )) dt, γ : [0 , → M, γ (0) = p, γ (1) = q (cid:27) . Similarly to the standard integrable case, B.Kruglikov (see [19]) proved that the integrated pseudodis-tance of the Kobayashi pseudometric coincides with the Kobayashi pseudodistance defined by chains ofpseudholomorphic discs. Then we define :
Definition 3.2. (1) The manifold ( M, J ) is Kobayashi hyperbolic if the integrated pseudodistance d ( M,J ) is a distance.(2) The manifold ( M, J ) is local Kobayashi hyperbolic at p ∈ M if there exist a neighborhood U of p and a positive constant C such that K ( M,J ) ( q, v ) ≥ C k v k for every q ∈ U and every v ∈ T q M .(3) A Kobayashi hyperbolic manifold ( M, J ) is complete hyperbolic if it is complete for the distance d ( M,J ) .3.2. Hyperbolicity of pseudoconvex regions of finite D’Angelo type.
In order to localize pseudoholo-morphic discs, we need the following technical Lemma (see [12] for a proof).
Lemma 3.3.
Let < r < and let θ r be a smooth nondecreasing function on R + such that θ r ( s ) = s for s ≤ r/ and θ r ( s ) = 1 for s ≥ r/ . Let ( M, J ) be an almost complex manifold, and let p be apoint of M . Then there exist a neighborhood U of p , positive constants A = A ( r ) ≥ , B = B ( r ) , and adiffeomorphism z : U → B such that z ( p ) = 0 , z ∗ J ( p ) = J st and the function log (cid:0) θ r (cid:0) | z | (cid:1)(cid:1) + θ r ( A | z | )+ B | z | is J -plurisubharmonic on U . In the next Proposition we give a priori estimates and a localization principle of the Kobayashi pseudo-metric. This proves the local Kobayashi hyperbolicity of J -pseudoconvex C regions of finite D’Angelotype. If ( M, J ) admits a global J -plurisubharmonic function, then K.Diederich and A.Sukhov proved in [9]the (global) Kobayashi hyperbolicity of a relatively compact J -pseudoconvex domains (with C boundary)by constructing a bounded strictly J -plurisubharmonic exhaustion function. We notice that, in our case,if the manifold ( M, J ) admits a global J -plurisubharmonic function then J -pseudoconvex C relativelycompact regions of finite D’Angelo type are also (globally) Kobayashi hyperbolic. Proposition 3.4.
Let D = { ρ < } be a domain of finite D’Angelo type in an almost complex manifold ( M, J ) , where ρ is a C defining function of D , J -plurisubharmonic in a neighborhood of D . Let p ∈ ¯ D and let U be a neighborhood of p in M . Then there exist positive constants C and s , and a neighborhood V ⊂ U of p in M , such that for each q ∈ D ∩ V and each v ∈ T q M : (3.1) K ( D,J ) ( q, v ) ≥ C k v k , (3.2) K ( D,J ) ( q, v ) ≥ sK ( D ∩ U,J ) ( q, v ) . This Proposition is a classical application of Lemma 3.3. This is due to N.Sibony [22] (see also [2] and[12] for a proof). For convenience we give the proof.
Proof.
According to Theorem 2.6, there exists a local peak J -plurisubharmonic function ϕ at p for D . Wecan choose constants < α < α ′ < β ′ < β and N > such that ϕ ≥ − β /N on {k z k < α } and ϕ ≤ − β /N on D ∩ { α ′ ≤ k z k ≤ β ′ } .We define ˜ ϕ by: ˜ ϕ := max (cid:0) N ϕ + k z k − β , − β (cid:1) if z ∈ D ∩ {k z k ≤ β ′ } , − β on D \{k z k ≤ β ′ } . The function k z k is J -plurisubharmonic on { q ∈ U : | z ( q ) | < } if k z ∗ J − J st k C ( B ) is sufficiently small.Then it follows that ˜ ϕ is J -plurisubharmonic on D. We may also suppose that ˜ ϕ is negative on D. Moreoverthe function ˜ ϕ − k z k is J -plurisubharmonic on D ∩ { q ∈ U : | z ( q ) | ≤ α } . Let θ α be a smooth non decreasing function on R + such that θ α ( s ) = s for s ≤ α / and θ α ( s ) = 1 for s ≥ α / . Set V = { q ∈ U : | z ( q ) | ≤ α } . According to Lemma 3.3, there are uniform positiveconstants A ≥ and B such that the function log (cid:0) θ α (cid:0) | z − z ( q ) | (cid:1)(cid:1) + θ α ( A | z − z ( q ) | ) + B k z k is J -plurisubharmonic on U for every q ∈ D ∩ V .We define for each q ∈ D ∩ V the function: Ψ q := θ α (cid:0) | z − z ( q ) | (cid:1) exp ( θ α ( A | z − z ( q ) | )) exp ( B ˜ ϕ ( z )) on D ∩ {k z k < α } , exp (1 + B ˜ ϕ ) on D \ {k z k < α } . The function logΨ q is J -plurisubharmonic on D ∩ {k z k < α } and, on D \ {k z k < α } , it coincides with B ˜ ϕ which is J -plurisubharmonic. Finally logΨ q is J -plurisubharmonic on the whole domain D .Let q ∈ V and let v ∈ T q M and consider a J -holomorphic disc u : ∆ → D such that u (0) = q and d u ( ∂/∂x ) = rv where r > . For ζ sufficiently close to 0 we have u ( ζ ) = q + d u ( ζ ) + O (cid:0) | ζ | (cid:1) . We define the following function φ ( ζ ) := Ψ q ( u ( ζ )) | ζ | which is subharmonic on ∆ \{ } since log φ is subharmonic. If ζ close to , then(3.3) φ ( ζ ) = | u ( ζ ) − q | | ζ | exp ( A | u ( ζ ) − q | ) exp ( B ˜ ϕ ( u ( ζ ))) . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 17
Setting ζ = ζ + iζ and using the J -holomorphy condition d u ◦ J st = J ◦ d u , we may write : d u ( ζ ) = ζ d u ( ∂/∂x ) + ζ J ( d u ( ∂/∂x )) . (3.4) | d u ( ζ ) | ≤ | ζ | ( k I + J k k d u ( ∂/∂x ) k ) According to (3.3) and to (3.4), we obtain that lim sup ζ → φ ( ζ ) is finite. Moreover setting ζ = 0 we have lim sup ζ → φ ( ζ ) ≥ k d u ( ∂/∂x ) k exp ( B ˜ ϕ ( q )) . Applying the maximum principle to a subharmonic extension of φ on ∆ we obtain the inequality k d u ( ∂/∂x ) k ≤ exp (1 − B ˜ ϕ ( q )) . Hence, by definition of the Kobayashi pseudometric, we obtain for every q ∈ D ∩ V and every v ∈ T q M : K ( D,J ) ( q, v ) ≥ (exp ( − B ˜ ϕ ( q ))) k v k . This gives estimate (3.1).Now in order to obtain estimate (3.2), we prove that there is a neighborhood V ⊂ U and a positiveconstant s such that for any J -holomorphic disc u : ∆ → D with u (0) ∈ V then u (∆ s ) ⊂ D ∩ U. Supposethis is not the case. We obtain a sequence ζ ν of ∆ and a sequence of J -holomorphic discs u ν such that ζ ν converges to 0, u ν (0) converges to p and k u ν ( ζ ν ) k / ∈ D ∩ U for every ν. According to the estimate (3.1),we obtain for a positive constant c > : c ≤ d ( D,J ) ( u ν (0) , u ν ( ζ ν )) ≤ d ∆ ( ζ ν , . This contradicts the fact that ζ ν converges to 0. (cid:3) The (global) Kobayahsi hyperbolicity is provided if we suppose that there is a global strictly J -plurisubharmonic function on ( M, J ) . Corollary 3.5.
Let D = { ρ < } be a relatively compact domain of finite D’Angelo type in an almostcomplex manifold ( M, J ) of dimension four, ρ being a defining function of D , J -plurisubharmonic in aneighborhood of D . Assume that ( M, J ) admits a global strictly J -plurisubharmonic function. Then ( D, J ) is Kobayahsi hyperbolic. As an application of the a priori estimate (3.1) of Proposition 3.4, we prove the tautness of D . Corollary 3.6.
Let D = { ρ < } be a relatively compact domain of finite D’Angelo type in an almostcomplex manifold ( M, J ) of dimension two. Assume that ρ is J -plurisubharmonic in a neighborhood of D .Moreover suppose that ( M, J ) admits a global strictly J -plurisubharmonic function. Then D is taut.Proof. Let ( u ν ) ν be a sequence of J -holomorphic discs in D . According to Corollary 3.5 the domain D ishyperbolic. Thus the sequence ( u ν ) ν is equiconituous, and then by Ascoli Theorem, we can extract fromthis sequence a subsequence still denoted ( u ν ) ν which converges to a map u : ∆ → D . Passing to thelimit the equation of J -holomorphicity of each u ν , it follows that u is a J -holomorphic disc. Since ρ is J -plurisubharmonic defining function for D , we have, by applying the maximun principle to ρ ◦ u , thealternative: either u (∆) ⊂ D or u (∆) ⊂ ∂D . (cid:3) We point out that the tautness of the domain D was proved, using a diferent method, by K.Diederich-A.Sukhov in [9]. Uniform estimates of the Kobayashi pseudometric.
In order to obtain more precise estimates, weneed to uniform estimates (3.1) of the Kobayashi pseudometric for a sequence of domains.
Proposition 3.7.
Assume that D = {ℜ ez + P ( z , z ) < } is a J st -pseudoconvex region of R , where P is a homogeneous polynomial of degree k ≤ m admitting a nonharmonic part. Let D ν be a sequence of J ν -pseudoconvex region of R such that ∈ ∂D ν is a boundary point of finite D’Angelo type l ν ≤ m .Suppose that D ν converges in the sense of local Hausdorff set convergence to D when ν tends to + ∞ andthat J ν converges to J st in the C topology when ν tends to + ∞ . Then there exist a positive constant C and a neighborhood V ⊂ U of the origin in R , such that for large ν and for every q ∈ D ν ∩ V and every v ∈ T q R K ( D ν ,J ) ( q, v ) ≥ C k v k . Proof.
Under the conditions of Proposition 3.7 we have the following Lemma:
Lemma 3.8.
For every large ν , there exists a diffeomorphism Φ ν : R → R with the following property: (1) The map ζ ( ζ, is a (Φ ν ) ∗ J ν -holomorphic disc of maximal contact order l ν . (2) The almost complex structure (Φ ν ) ∗ J ν satisfies conditions (2.2) and (2.3). (3) Φ ν ( D ν ) = { ρ ν < } with ρ ν = ℜ ez + m X j =2 l ν P j,ν ( z , z ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) < , where P j,ν are homogeneous polynomials of degree j and P l ν ,ν contains a nonharmonic part de-noted by P ∗ l ν ,ν = 0 . (4) we have inf ν {k P l ν ,ν k} > . Moreover the sequence of diffeomorphisms Φ ν converges to the identity on any compact subsets of R in the C topology. The crucial fact used to prove Proposition 3.7 is the point (4) , which is a direct consequence of theconvergence of Φ ν ( D ν ) to D . Hence the proof of Proposition 3.7 is similar to Theorem 2.6 and Theorem3.4, where all the constants are uniform. (cid:3) H¨older extension of diffeomorphisms.
This subsection is devoted to the boundary continuity of dif-feomorphisms. This is stated as follows:
Proposition 3.9.
Let D = { ρ < } and D ′ = { ρ ′ < } be two relatively compact domains of finiteD’Angelo type m in four dimensional almost complex manifolds ( M, J ) and ( M ′ , J ′ ) . We suppose that ρ (resp. ρ ′ ) is a J (resp J ′ )-plurisubharmonic defining function on a neighborhood of D (resp. D ′ ). Let f : D → D ′ be a ( J, J ′ ) -biholomorphism. Then f extends as a H¨older homeomorphism with exponent / m between D and D ′ . Estimates of the Kobayashi pseudometric obtained by H.Gaussier and A.Sukhov in [12] provide theH ¨older extension with exponent / up to the boundary of a biholomorphism between two strictly pseu-doconvex domains (see Proposition 3.3 of [6]). Similarly, in order to obtain Proposition 3.9, we begin byestablishing a more precise estimate than (3.1) of Proposition 3.4. Proposition 3.10.
Let D = { ρ < } be a domain of finite D’Angelo type in a four dimensional almostcomplex manifold ( M, J ) , where ρ is a C defining function of D , J -plurisubharmonic in a neighborhood SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 19 of D . Let p ∈ ∂D and let U be a neighborhood of p in M . Then there are positive constant C and aneighborhood V ⊂ U of p in M , such that for every q ∈ D ∩ V and every v ∈ T q M : (3.5) K ( D,J ) ( q, v ) ≥ C k v k dist ( q, ∂D ) / m . Proof of Proposition 3.10.
Let p ∈ ∂D . We may suppose that D ⊂ R , p = 0 and that J satisfies (2.2)and (2.3). Let q ′ be a boundary point in a neighborhood of the origin and let ϕ q ′ be the local peak J -plurisubharmonic function at q ′ given by Theorem 2.6. There are positive constants C and C such that(3.6) − C k z − q ′ k ≤ ϕ q ′ ( z ) ≤ − C Ψ q ′ ( z ) , where Ψ q ′ ( z ) := | z − q ′ | m + | z − q ′ | + | z − q ′ | | z − q ′ | is a J -plurisubharmonic function on a neighborhood U of the origin.Now consider a J -holomorphic disc u : ∆ → D , such that u (0) is sufficiently close to the origin andthen, according to Proposition 3.4, we have u (∆ s ) ⊂ D ∩ U, for some < s < depending only on u (0) .We assume that q ′ is such that dist ( u (0) , ∂D ) = k u (0) − q ′ k . According to the J -plurisubharmonicity of Ψ q ′ , we have for | ζ | ≤ s : Ψ q ′ ( u ( ζ )) ≤ C π Z π Ψ q ′ (cid:16) u (cid:16) re iθ (cid:17)(cid:17) dθ, for some positive constant C . Hence using (3.6) and the J -plurisubharmonicity of ϕ q ′ we obtain: Ψ q ′ ( u ( ζ )) ≤ − C πC Z π ϕ q ′ (cid:16) u (cid:16) re iθ (cid:17)(cid:17) dθ ≤ − C C ϕ q ′ ( u (0)) . Since there is a positive constant C such that k u ( ζ ) − q ′ k m ≤ C Ψ q ′ ( u ( ζ )) and using (3.6), we finally obtain: k u ( ζ ) − q ′ k m ≤ C C C C dist ( u (0) , ∂D ) . Hence there exists a positive constant C such that: dist ( u ( ζ ) , ∂D ) ≤ C dist ( u (0) , ∂D ) / m , whenever ζ ≤ s .According to Lemma . of [17] there is a positive constant C such that: k∇ u (0) k ≤ C sup | ζ |
Lemma 3.11.
Let D be a domain in an almost complex manifold ( M, J ) . Then there is a positive constant C such that for any p ∈ D and any v ∈ T p M : (3.7) K ( D,J ) ( p, v ) ≤ C k v k dist ( p, ∂D ) . Lemma 3.12. (Hopf lemma) Let D be a relatively compact domain with a C boundary on an almostcomplex manifold ( M, J ) . Then for any negative J -plurisubharmonic function ρ on D there exists a constant C > such that for any p ∈ D : | ρ ( p ) | ≥ C dist( p, ∂D ) . Now we can go on the proof of Proposition 3.9.
Proof of Proposition 3.9.
Let f : D → D ′ be a ( J, J ′ ) -biholomorphism. According to Proposition 3.10 andto the decreasing property of the Kobayashi pseudometric there is a positive constant C such that for every p ∈ D sufficiently close to the boundary and every v ∈ T p MC k d p f ( v ) k dist ( f ( p ) , ∂D ′ ) m ≤ K ( D ′ ,J ′ ) ( f ( p ) , d p f ( v )) = K ( D,J ) ( p, v ) . Due to Lemma 3.11 there exists a positive constant C such that: K ( D,J ) ( p, v ) ≤ C k v k dist ( p, ∂D ) . This leads to: k d p f ( v ) k ≤ C C dist ( f ( p ) , ∂D ′ ) m dist ( p, ∂D ) k v k . Moreover the Hopf lemma 3.12 for almost complex manifolds applied to ρ ′ ◦ f and ρ ◦ f − and the fact that ρ and ρ ′ are defining functions, provides the following boundary distance preserving property: C dist ( p, ∂D ) ≤ dist (cid:0) f ( p ) , ∂D ′ (cid:1) ≤ C dist ( p, ∂D ) , for some positive constat C . Finally this implies: k d p f ( v ) k ≤ C C C k v k dist ( p, ∂D ) m − m . This gives the desired statement. (cid:3)
4. S
HARP ESTIMATES OF THE K OBAYASHI PSEUDOMETRIC
In this section we give sharp lower estimates of the Kobayashi pseudometric in a pseudoconvex regionnear a boundary point of finite D’Angelo type less than or equal to four. This condition will appear necessary,in our proof, as explained in the appendix. Moreover in order to give sharp estimates near a point of arbitraryfinite D’Angelo type, we are also interested in the nontangential behaviour of the Kobayashi pseudometric.The main result of this section is the following theorem (see also Theorem B):
Theorem 4.1.
Let D = { ρ < } be a relatively compact domain of finite D’Angelo type less than or equalto four in an almost complex manifold ( M, J ) of dimension four, where ρ is a C defining function of D , J -plurisubharmonic on a neighborhood of D . Then there exists a positive constant C with the followingproperty: for every p ∈ D and every v ∈ T p M there is a diffeomophism, Φ p ∗ , in a neighborhood U of p ,such that: (4.1) K ( D,J ) ( p, v ) ≥ C (cid:18) | ( d p Φ p ∗ v ) | τ ( p ∗ , | ρ ( p ) | ) + | ( d p Φ p ∗ v ) || ρ ( p ) | (cid:19) , where τ ( p ∗ , | ρ ( p ) | ) is defined by (4.3). SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 21
As a direct consequence we have: (4.2) K ( D,J ) ( p, v ) ≥ C ′ | ( d p Φ p ∗ v ) || ρ ( p ) | + | ( d p Φ p ∗ v ) || ρ ( p ) | ! , for a positive constant C ′ . In complex manifolds, D.Catlin [5] first obtained such an estimate, based on lower estimates of theCarath´eodory pseudometric. F.Berteloot [3] gave a different proof based on a Bloch principle. Our proofwich is inspired by the proof of F.Berteloot is based on some scaling method.4.1.
The scaling method.
We consider here a pseudoconvex region D = { ρ < } of finite D’Angelo type m in R , where ρ has the following expression on a neighborhood U of the origin: ρ ( z , z ) = ℜ ez + H m ( z , z ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) . where H m is a homogeneous subharmonic polynomial of degree m admitting a nonharmonic part.Assume that p ν is a sequence of points in D ∩ U converging to the origin. For each p ν sufficiently closeto ∂D , there exists a unique point p ∗ ν ∈ ∂D ∩ U such that p ∗ ν = p ν + (0 , δ ν ) , with δ ν > . Notice that for large ν , the quantity δ ν is equivalent to dist ( p ν , ∂D ∩ U ) and to | ρ ( p ν ) | .We consider a diffeomorphism Φ ν : R → R satisfying:(1) Φ ν ( p ∗ ν ) = 0 and Φ ν ( p ν ) = (0 , − δ ν ) .(2) Φ ν converges to Id : R → R on any compact subset of R in the C sense.(3) When we denote by D ν := Φ ν ( D ∩ U ) which admits the defining function is ρ ν := ρ ◦ (Φ ν ) − and by J ν := (Φ ν ) ∗ J , then ρ ν is given by: ρ ν ( z , z ) = ℜ ez + m X k =2 l ν P k ( z , z , p ∗ ν ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) , where the polynomial P l ν contains a nonharmonic part. Moreover J ν satisfies (2.2) and (2.3).This is done by considering first the translation T ν of R given by z z − p ∗ ν . According to J.-F.Barraudand E.Mazzilli [1] that the D’Angelo type is an upper semicontinuous function in a four dimensional almostcomplex manifold. Thus the D’Angelo type of points in a small enough neighborhood can only be smallerthan at the point itself. Then we consider a ( T ν ) ∗ J -holomorphic disc u of maximal contact order l ν ,where l ν ≤ m is the D’Angelo type of p ∗ ν . We choose coordinates such that u is given by u ( ζ ) =( ζ, , and such that ( T ν ) ∗ J ( z ,
0) = J st and T ( ∂T ν ( D )) ∩ J (0) T ( ∂T ν ( D )) = { z = 0 } . Then byconsidering the family of vectors (1 , at base points (0 , t ) for t = 0 small enough, we obtain a familyof pseudoholomorphic discs u t such that u t (0) = (0 , t ) and d u t ( ∂/∂ x ) = (0 , . Due to the parametersdependance of the solution to the J ν -holomorphy equation, we straighten these discs into the lines { z = t } .Next we consider a transversal foliation by pseudoholomorphic discs passing through ( t, and ( t, − δ ν ) for t small enough and we straighten these lines into { z = c } . This leads to the desired diffeomorphism Φ ν of R .Now, we need to remove harmonic terms from the polynomial m − X k =2 l ν P k ( z , z , p ∗ ν ) . So we consider a biholomorphism (for the standard structure) of C with the following form: ϕ ν ( z , z ) := z , z + m − X k =2 l ν ℜ e (cid:16) c k,ν z k (cid:17) , where c k,ν are well chosen complex numbers. Then the diffeomorphism Φ ν := ϕ ν ◦ Φ ν satisfies:(1) Φ ν ( p ∗ ν ) = 0 and Φ ν ( p ν ) = (0 , − δ ν ) .(2) Φ ν converges to Id : R → R on any compact subset of R in the C sense.(3) If we denote by D ν := Φ ν ( D ∩ U ) the domain with the defining function ρ ν := ρ ◦ (Φ ν ) − , then ρ ν is given by: ρ ν ( z , z ) = ℜ ez + m − X k =2 l ν P ∗ k ( z , z , p ∗ ν ) + P m ( z , z , p ∗ ν ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) , where the polynomial m − X k =2 l ν P ∗ k ( z , z , p ∗ ν ) does not contain any harmonic terms. Moreover the polynomial P ∗ l ν is not idencally zero. More-over, generically, J ν := (Φ ν ) ∗ J is no more diagonal.Since the origin is a boundary point of D’Angelo type m for D , it follows that, denoting by P ∗ m thenonharmonic part of P m , we have P ∗ m ( .,
0) = H ∗ m = 0 , where H ∗ m is the nonharmonic part of H m .This allows to define for large ν :(4.3) τ ( p ∗ ν , δ ν ) := min k =2 l ν , ··· , m (cid:18) δ ν k P ∗ k ( ., p ∗ ν ) k (cid:19) k . Moreover the following inequalities hold:(4.4) C δ ν ≤ τ ( p ∗ ν , δ ν ) ≤ Cδ m ν , where C is a positive constant. The right inequality comes from the fact that k P ∗ m ( ., p ∗ ν ) k ≥ C > for large ν . And the left one comes the fact that there exists a positive constant C such that for every l ν ≤ k ≤ m we have k P ∗ k ( ., p ∗ ν ) k ≤ C .Now we consider the nonisotropic dilation Λ ν of C : Λ ν : ( z , z ) (cid:16) τ ( p ∗ ν , δ ν ) − z , δ − ν z (cid:17) . We set ˜ D ν := Λ ν ( D ν ) the domain admitting the defining function ˜ ρ ν := δ − ν ρ ν ◦ Λ − ν and ˜ J ν :=(Λ ν ) ∗ ( J ν ) the direct image of J ν under Λ ν .The next lemma is devoted to describe ( ˜ D ν , ˜ J ν ) when passing at the limit. Lemma 4.2. (1)
The domain ˜ D ν converges in the sense of local Hausdorff set convergence to a (standard) pseudo-convex domain ˜ D = { ˜ ρ < } , with ˜ ρ ( z ) = ℜ ez + P ( z , z ) , where P is a nonzero subharmonic polynomial of degree smaller than or equal to m which admitsa nonharmonic part. SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 23 (2)
In case the origin is of D’Angelo type four for D , the sequence of almost complex structures ˜ J ν converges on any compact subsets of C in the C sense to J st .Proof. We first prove part 1. Due to inequalities (4.4), the defining function of ˜ D ν satisfies: ˜ ρ ν = ℜ ez + m X k =2 l ν δ − ν τ ( p ∗ ν , δ ν ) k P ∗ k ( z , z , p ∗ ν ) + δ − ν τ ( p ∗ ν , δ ν ) m P m ( z , z , p ∗ ν ) + O ( τ ( δ ν )) . Passing to a subsequence, we may assume that the polynomial m X k =2 l ν δ − ν τ ( p ∗ ν , δ ν ) k P ∗ k ( z , z , p ∗ ν ) + δ − ν τ ( p ∗ ν , δ ν ) m P m ( z , z , p ∗ ν ) converges uniformly on compact subsets of C to a nonzero polynomial P of degree ≤ m admittinga nonharmonic part. Since the pseudoconvexity is invariant under diffeomorphisms, it follows that thedomains ˜ D ν are ˜ J ν -pseudoconvex, and then passing to the limit, the domain ˜ D is J st -pseudoconvex. Thusthe polynomial P is subharmonic.We next prove part 2. The complexification of the almost complex structure J ν is given by ( J ν ) C = X l =1 (cid:16) A l,l ( z ) dz l ⊗ ∂∂z l + B l,l ( z ) dz l ⊗ ∂∂z l + B l,l ( z ) dz l ⊗ ∂∂z l + A l,l ( z ) dz l ⊗ ∂∂z l (cid:17) + A , ( z ) dz ⊗ ∂∂z + B , ( z ) dz ⊗ ∂∂z + B , ( z ) dz ⊗ ∂∂z + A , ( z ) dz ⊗ ∂∂z , where A l,l ( z ) = i + O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + X k =2 c k,ν z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for l = 1 , ,B l,l ( z ) = O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + X k =2 c k,ν z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! for l = 1 , ,A , ( z ) = X k =2 kc k,ν z k − O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + X k =2 c k,ν z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,B , ( z ) = X k =2 k (cid:16) c k,ν z k − − c k,ν z k − (cid:17) O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + X k =2 c k,ν z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! . By a direct computation, the complexification of ˜ J ν is equal to: (cid:16) ˜ J ν (cid:17) C = X l =1 ( A l,l (Λ − ν ( z )) dz l ⊗ ∂∂z l + B l,l (Λ − ν ( z )) dz l ⊗ ∂∂z l + B l,l (Λ − ν ( z )) dz l ⊗ ∂∂z l + A l,l (Λ − ν ( z )) dz l ⊗ ∂∂z l ) + τ ( p ∗ ν , δ ν ) δ − ν A , (Λ − ν ( z )) dz ⊗ ∂∂z + τ ( p ∗ ν , δ ν ) δ − ν B , (Λ − ν ( z )) dz ⊗ ∂∂z + τ ( p ∗ ν , δ ν ) δ − ν B , (Λ − ν ( z )) dz ⊗ ∂∂z + τ ( p ∗ ν , δ ν ) δ − ν A , (Λ − ν ( z )) dz ⊗ ∂∂z . According to (4.4) and since c k,ν converges to zero when ν tends to + ∞ for k = 2 , , it follows that ˜ J ν converges to J st . This proves part (2) . (cid:3) Complete hyperbolicity in D’Angelo type four condition.
In this subsection we prove Theorem 4.1.Keeping notations of the previous subsection; we start by establishing the following lemma which gives aprecise localization of pseudoholomorphic discs in boxes.
Lemma 4.3.
Assume the origin ∈ ∂D is a point of D’Angelo type four. There are positive constants C , δ and r such that for any < δ < δ , for any large ν and for any J ν -holomorphic disc g ν : ∆ → D ν wehave : g ν (0) = (0 , − δ ν ) ⇒ g ν ( r ∆) ⊂ Q (0 , C δ ν ) , where Q (0 , δ ν ) := { z ∈ C : | z | ≤ τ ( p ∗ ν , δ ν ) , | z | ≤ δ ν } .Proof. Proof of Lemma 4.3 . Assume by contradiction that there are a sequence ( C ν ) ν that tends to + ∞ as ζ ν converges to in ∆ , and J ν -holomorphic discs g ν : ∆ → D ν such that g ν (0) = (0 , − δ ν ) and g ν ( ζ ν ) Q (0 , C ν δ ν ) . We consider the nonisotropic dilations of C : Λ rν : ( z , z ) (cid:16) r τ ( p ∗ ν , δ ν ) − z , rδ − ν z (cid:17) , where r is a positive constant to be fixed. We set h ν := Λ rν ◦ g ν , ˜ ρ rν := rδ − ν ρ ν ◦ (Λ rν ) − and ˜ J rν := (Λ rν ) ∗ J ν .It follows from Lemma 4.2 that ˜ ρ rν converges to ˜ ρ = Re ( z ) + P ( z , z ) uniformly on any compact subset of C and ˜ J rν converges to J st , uniformly on any compact subset of C .According to the stability of the Kobayashi pseudometric stated in Proposition 3.7, there exist a positiveconstant C and a neighborhood V of the origin in R , such that for every large ν , for every q ∈ ˜ D ν ∩ V andevery v ∈ T q R : K ( ˜ D ν , ˜ J ν ) ( q, v ) ≥ C k v k . Therefore, there exists a constant C ′ > such that k dh ν ( ζ ) k≤ C ′ for any ζ ∈ (1 /
2) ∆ satisfying h ν ( ζ ) ∈ ˜ D ν ∩ V ′ , with V ′ ⊂ V . Now we choose the constant r such that h ν (0) = (0 , − r ) ∈ Int ( V ′ ) . On the other hand, the sequence | h ν ( ζ ν ) | tends to + ∞ . Denote by [0 , ζ ν ] thesegment (in C ) joining the origin and ζ ν and let ζ ′ ν = r ν e iθ ν ∈ [0 , ζ ν ] be the point closest to the origin suchthat h ν ([0 , ζ ′ ν ]) ⊂ ˜ D ν ∩ V and h ν ( ζ ′ ν ) ∈ ∂V . Since h ν (0) ∈ Int ( V ′ ) , we have k h ν (0) − h ν (cid:0) ζ ′ ν (cid:1) k ≥ C ′′ SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 25 for some constant C ′′ > . It follows that: k h ν (0) − h ν (cid:0) ζ ′ ν (cid:1) k ≤ Z r ν (cid:13)(cid:13)(cid:13) dh ν (cid:16) te iθ ν (cid:17)(cid:13)(cid:13)(cid:13) dt ≤ C ′ r ν −→ . This contradiction proves Lemma 4.3. (cid:3)
Now we go on the proof of Theorem 4.1.
Proof of Theorem 4.1.
Due to the localization of the Kobayashi pseudometric established in Proposition 3.4,it suffices to prove Theorem 4.1 in a neighborhood U of q ∈ ∂D . Choosing local coordinates z : U → B ⊂ R centered at q , we may assume that D ∩ U = { ρ < } is a J -pseudonconvex region of ( R , J ) ,that q = 0 ∈ ∂D and that J satisfies (2.2) and (2.3). We also suppose that the complex tangent space T ∂D ∩ J (0) T ∂D at of ∂D is given by { z = 0 } . Moreover the defining function ρ is expressed by: ρ ( z ) = ℜ ez + H m ( z , z ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) For p ∈ D ∩ U be sufficiently close to the boundary ∂D , there exists a unique point p ∗ ∈ ∂D ∩ U suchthat p ∗ = p + (0 , δ ) , with δ > . We define an infinitesimal pseudometric N on D ∩ U ⊆ R by:(4.5) N ( p, v ) := | ( d p Φ p ∗ v ) | τ ( p ∗ , | ρ ( p ) | ) + | ( d p Φ p ∗ v ) || ρ ( p ) | , for every p ∈ D ∩ U and every v ∈ T p R , where Φ p ∗ is defined as diffeomorphisms Φ ν (of previoussubsection) for p ∗ instead of p ∗ ν .To prove estimate (4.1) of Theorem 4.1, it suffices to find a positive constant C such that for any J -holomorphic disc u : ∆ → D ∩ U , we have:(4.6) N ( u (0) , d u ( ∂/∂ x )) ≤ C. Indeed, for a J -holomorphic disc u such that u (0) = p and d u ( ∂/∂ x ) = rv , (4.6) leads to r = N ( p, v ) N ( u (0) , d u ( ∂/∂ x )) ≥ N ( p, v ) C .
Suppose by contradiction that (4.6) is not true, that is, there is a sequence of J -holomorphic discs u ν :∆ → D ∩ U such that N ( u ν (0) , d u ν ( ∂/∂ x )) ≥ ν . Then we consider a sequence ( y ν ) ν of points in ∆ / such that:(1) | y ν | ≤ νN ( u ν ( y ν ) , d y ν u ν ( ∂/∂x )) ,(2) N ( u ν ( y ν ) , d y ν u ν ( ∂/∂x )) ≥ ν , and(3) y ν + ∆ ν/N ( u ν ( y ν ) ,d yν u ν ( ∂/∂x )) ⊆ ∆ / for sufficiently large ν .This allows to define a sequence of J -holomorphic discs g ν : ∆ ν → D ∩ U by g ν ( ζ ) := u ν (cid:18) y ν + ζ N ( u ν ( y ν ) , d y ν u ν ( ∂/∂x )) (cid:19) . Consider the sequence g ν = u ν ( y ν ) in D ∩ U . Since | y ν | ≤ /ν and since the C norm of any J -holomorphicdisc u ν is uniformally bounded it follows that g ν (0) converges to the origin.We apply the scaling method to the sequence g ν (0) . We denote by g ν (0) ∗ the boundary point given by g ν (0) ∗ := g ν (0) + (0 , δ ν ) . We set the scaled disc ˜ g ν := Λ ν ◦ Φ ν ◦ g ν , where diffeomorphisms Λ ν and Φ ν are define in the subsection about the scaling method. In order to extract from ˜ g ν a subsequence whichconverges to a Brody curve, we need the following Lemma. Lemma 4.4.
There is a positive constant r such that: (1) There exists a positive constant C such that (4.7) ˜ g ν ( r ∆ ν ) ⊂ ∆ C × ∆ C . (2) There exists a positive constant C such that for every large ν we have : (4.8) k d ˜ g ν k C ( r ∆ ν ) ≤ C . Proof.
We prove the first part. We define a J ν -holomorphic disc h ν ( ζ ) := Φ ν ◦ g ν ( νζ ) from the unit disc ∆ to D ν . According to Lemma 4.3, since h ν (0) = Φ ν ◦ g ν (0) = (0 , − δ ν ) , we have h ν ( r ∆) ⊆ Q (0 , C δ ν ) for some positive constants r and C . Hence Φ ν ◦ g ν ( r ∆ ν ) ⊆ Q (0 , C δ ν ) . After dilations, this leads to (4.7).Then we prove the second part. According to Lemma 4.2, the sequence of almost complex structures ˜ J ν converges on any compact subsets of C in the C sense to J st . Then for sufficiently large ν , the norm k ˜ J ν − J st k C (∆ C × ∆ C ) is as small as necessary. So for large ν , and due to Proposition . . of J.-C.Sikoravin [23] there exists C > such that (4.8) holds. (cid:3) Hence according to Lemmas 4.2 and 4.4 we may extract from ˜ g ν a subsequence, still denoted by ˜ g ν wichconverges in C topology to a standard complex line ˜ g : C → ( { Rez + P ( z , z ) < } , J st ) . The polynomial P is subharmonic and contains a nonharmonic part; this implies that the domain ( { Rez + P ( z , z ) < } , J st ) is Brody hyperbolic and so the complex line ˜ g is constant. To obtain acontradiction, we prove that the derivative of ˜ g at the origin is nonzero:
12 = N ( g ν (0) , d g ν ( ∂/∂ x )) = | ( d (Φ ν ◦ g ν ) ( ∂/∂ x )) | τ ( g ν (0) ∗ , | ρ ( g ν (0)) | ) + | ( d (Φ ν ◦ g ν ) ( ∂/∂ x )) || ρ ( g ν (0)) | . Since | ρ ( g ν (0)) | is equivalent to δ ν , it follows that for some positive constant C and for large ν , we have: ≤ C (cid:18) | ( d (Φ ν ◦ g ν ) ( ∂/∂ x )) | τ ( g ν (0) ∗ , δ ν ) + | ( d (Φ ν ◦ g ν ) ( ∂/∂ x )) | δ ν (cid:19) = C k d ˜ g ν ( ∂/∂ x ) k . Since ˜ g ν converges to ˜ g in the C sense, it follows that d ˜ g ( ∂/∂ x ) = 0 , providing a contradiction. Thisachieves the proof of Theorem 4.1. (cid:3) Estimate (4.2) of the Kobayashi pseudometric allows to study the completness of the Kobayashi pseu-dodistance D . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 27
Corollary 4.5.
Let D = { ρ < } be a relatively compact domain of finite D’Angelo type less than or equalto four in an almost complex manifold ( M, J ) of dimension four, where ρ is a defining function of D , J -plurisubharmonic in a neighborhood of D . Assume that ( M, J ) admits a global strictly J -plurisubharmonicfunction. Then ( D, J ) is complete hyperbolic.Proof. The fact that ( M, J ) admits a global strictly J -plurisubharmonic function and estimate (3.1) ofProposition 3.4 leads to the Kobayashi hyperbolicity of D . Then estimate (4.2) of the Kobayashi pseudo-metric stated in Theorem 4.1 gives the completness of the metric space (cid:0) D, d ( D,J ) (cid:1) by a classical integrationargument. (cid:3) Regions with noncompact automorphisms group.
The next corollary is devoted to regions withnoncompact automorphisms group.
Corollary 4.6.
Let D = { ρ < } be a relatively compact domain in a four dimensional almost complexmanifold ( M, J ) of finite D’Angelo type less than or equal to four. Assume that ρ is a C defining function of D , J -plurisubharmonic on a neighborhood of D . If there is an automorphism of D with orbit accumulatingat a boundary point then there exists a polynomial P of degree at most four, without harmonic part such that ( D, J ) is biholomorphic to ( {ℜ ez + P ( z , z ) < } , J st ) . If the domain D is a relatively compact strictly J -pseudoconvex domain with noncompact automorphismsgroup then ( D, J ) is biholomorphic to a model domain. This was proved by H.Gaussier and A.Sukhov in[12] in dimension four and by K.H.Lee in [20] in arbitrary (even) dimension. Sketch of the proof.
We suppose that for some point p ∈ D , there is a sequence f ν of automorphisms of ( D, J ) such that p ν := f ν ( p ) converges to ∈ ∂D . We apply the scaling method to the sequence p ν . Stillkeeping notations of subsection . , we set F ν := Λ ν ◦ Φ ν ◦ f ν : f − ν ( D ∩ U ) → ˜ D ν . This sequence of biholomorphisms is such that :(1) (cid:0) f − ν ( D ∩ U ) (cid:1) ν converges to D .(2) ˜ D ν converges to a pseudoconvex domain ˜ D = { Rez + P ( z , z ) < } , where P is a nonzero sub-harmonic polynomial of degree ≤ which contains a nonharmonic part. Changing ˜ D by applying astandard biholomorphism if necessary, we may suppose that P ( z , z ) is without harmonic terms.(3) For any compact subset K ⊂ D , the sequence (cid:0) k F ν k C ( K ) (cid:1) ν is bounded.Hence, we may extract from ( F ν ) ν a subsequence converging, on any compact subset of D in the C ∞ sense,to a ( J, J st ) -holomorphic map F : D −→ ¯˜ D . Finally F is a ( J, J st ) -biholomorphism from D to ˜ D . (cid:3) Nontangential approach in the general setting.
In this subsection, refering to I.Graham [14], wegive a sharp estimate of the Kobayashi pseudometric of a pseudoconvex region in a cone with vertex at aboundary point of arbitrary finite D’Angelo type. We denote by
Λ := {−ℜ ez > k k z k} , where < k < ,the cone with vertex at the origin and axis the negative real z axis. Theorem 4.7.
Let D = { ρ < } be a domain of finite D’Angelo type in (cid:0) R , J (cid:1) , where ρ ( z , z ) = ℜ ez + H m ( z , z ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) , is a C defining function of D , J -plurisubharmonic on a neighborhood of D . We suppose that H m is ahomogeneous subharmonic polynomial of degree m admitting a nonharmonic part. Then there exists apositive constant C such that for every p ∈ D ∩ Λ and every v ∈ T p M : K ( D,J ) ( p, v ) ≥ C | v || ρ ( p ) | m + | v || ρ ( p ) | ! . Before proving Theorem 4.7 we need the following crucial lemma.
Lemma 4.8.
There exist a neighborhood U of the origin and a positive constant C such that if p ∈ D ∩ U ∩ Λ then p ∈ n z ∈ C : | z | < C dist ( p, ∂D ) m , | z | < C dist ( p, ∂D ) o . Proof.
According to the fact that dist ( z, ∂D ) is equivalent to | ρ ( z ) | = −ℜ ez + O (cid:0) k z k (cid:1) and to thedefinition of the cone Λ , we have: lim z → ,z ∈ D ∩ Λ −ℜ ez dist ( z, ∂D ) = 1 . This implies the existence of a positive constant C such that k p k < − k ℜ ep ≤ C dist ( p, ∂D ) , whenever p ∈ D ∩ Λ is sufficiently close to the origin. Thus p ∈ n z ∈ C : | z | < C dist ( p, ∂D ) m , | z | < C dist ( p, ∂D ) o , for p ∈ D ∩ Λ sufficiently close to the origin. (cid:3) The proof of Theorem 4.7 is similar and easier than proof of Theorem 4.1. For convenience, we write it.
Proof of Theorem 4.7.
Let U be a neighborhood of the origin. We define an infinitesimal pseudometric N on D ∩ U ⊆ R by: N ( p, v ) := | v || ρ ( p ) | m + | v || ρ ( p ) | , for every p ∈ D ∩ U and every v ∈ T p C .We have to find a positive constant C such that for every J -holomorphic disc u : ∆ → D ∩ U , such thatif u (0) ∈ Λ then: N ( u (0) , d u ( ∂/∂ x )) ≤ C. Suppose by contradiction that this inequality is not true, that is, there exists a sequence of J -holomorphicdiscs u ν : ∆ → D ∩ U such that u ν (0) ∈ Λ and N ( u ν (0) , d u ν ( ∂/∂ x )) ≥ ν . Then consider a sequence ( y ν ) ν of points in ∆ / such that(1) | y ν | ≤ νN ( u ν ( y ν ) , d y ν u ν ( ∂/∂x )) ,(2) N ( u ν ( y ν ) , d y ν u ν ( ∂/∂x )) ≥ ν , and(3) y ν + ∆ ν/N ( u ν ( y ν ) ,d yν u ν ( ∂/∂x )) ⊆ ∆ / for sufficiently large ν .Then we define a sequence of J -holomorphic discs g ν : ∆ ν → D ∩ U by g ν ( ζ ) := u ν (cid:18) y ν + ζ N ( u ν ( y ν ) , d y ν u ν ( ∂/∂x )) (cid:19) . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 29
For large ν , we have g ν (0) = u ν ( y ν ) in D ∩ U ∩ Λ and g ν (0) converges to the origin. Set δ ν := dist ( g ν (0) , ∂D ) , and consider the following dilations of C : Λ ν : ( z , z ) (cid:18) δ − m ν z , δ − ν z (cid:19) . In order to extract from Λ ν ◦ g ν a subsequence which converges to a Brody curve, we need the followingLemma. Lemma 4.9.
There exists a positive constant r such that: (1) there exists a positive constant C such that: (4.9) Λ ν ◦ g ν ( r ∆ ν ) ⊂ ∆ C × ∆ C , (2) there is a positive constant C such that for every large ν we have : (4.10) k d (Λ ν ◦ g ν ) k C ( r ∆ ν ) ≤ C . Proof.
We first prove (4.9). We define a new J -holomorphic disc h ν ( ζ ) := g ν ( νζ ) from the unit disc ∆ to D ν . According to Lemma 4.8, we have h ν (0) = g ν (0) ∈ { z ∈ C : | z | ≤ C δ m ν , | z | ≤ C δ ν } . This implies: h ν ( r ∆) ⊆ { z ∈ C : | z | ≤ C δ m ν , | z | < C δ ν } , for positive constants r and C , since Lemma 4.3 is true if we replace τ ( p ∗ ν , δ ν ) by δ m ν . Hence g ν ( r ∆ ν ) ⊆ { z ∈ C : | z | < C δ m ν , | z | ≤ C δ ν } . After dilations, this leads to (4.9).The proof of (4.10) is similar to (4.8) of Lemma 4.4, since the sequence of structures (Λ ν ) ∗ J convergeson any compact subset of C in the C sense to J st because J is diagonal. (cid:3) Hence according to Lemma 4.9 we may extract from Λ ν ◦ g ν a subsequence, still denoted by Λ ν ◦ g ν wichconverges in the C sense to a standard complex line ˜ g : C → ( { Rez + H m ( z , z ) < } , J st ) , wherethe domain ( { Rez + P ( z , z ) < } , J st ) is Brody hyperbolic since H m ( z , z ) contains a nonharmonicpart. Then the standard complex line ˜ g is constant. To obtain a contradiction, we prove that the derivative of ˜ g is nonzero:
12 = N ( g ν (0) , d g ν ( ∂/∂ x )) = | ( d g ν ( ∂/∂ x )) || ρ ( g ν (0)) | m + | ( d g ν ( ∂/∂ x )) || ρ ( g ν (0)) | . Since | ρ ( g ν (0)) | is equivalent to δ ν , it follows that for some positive constant C we have for large ν : ≤ C | ( d ( g ν )( ∂/∂ x )) | δ m ν + | ( d ( g ν )( ∂/∂ x )) | δ ν ! = C k d (Λ ν ◦ g ν )( ∂/∂ x ) k . This provide a contradiction. (cid:3)
5. A
PPENDIX : C
ONVERGENCE OF THE STRUCTURES INVOLVED BY THE SCALING METHOD .In this appendix, we prove that, generically, the convergence of the sequence of structures involved by thescaling method to the standard structure J st occurs only on a neighborhood of boundary points of D’Angelotype less than or equal to four.Let D = { ρ < } be a pseudoconvex region of finite D’Angelo type m in R , where ρ has the followingexpression on a neighborhood U of the origin: ρ ( z , z ) = ℜ ez + H m ( z , z ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) , where H m is a homogeneous subharmonic polynomial of degree m admitting a nonharmonic part. As-sume that p ν is a sequence of points in D ∩ U converging to the origin, and, for large ν , consider thesequence of diffeomorphisms Φ ν : R → R given in the scaling method. We suppose that the function ρ ν = ρ ◦ (Φ ν ) − is given by: ρ ν ( z , z ) = ℜ ez + ℜ e (cid:0) α ν z (cid:1) + βν | z | + m X k =3 P k ( z , z , p ∗ ν ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) . Moreover the structure J ν := (Φ ν ) ∗ J satisfies (2.2) and (2.3). To fix notations, we set: J ν = a ν b ν c ν − a ν a ν b ν c ν − a ν . Now, consider the following diffeomorphism of R defined by:(5.1) Ψ − ν ( x , y , x , y ) = ( x + R ,ν , y + S ,ν , x + R ,ν , y + S ,ν ) converging to the identity and such that d Ψ − ν = Id . We suppose that R k,ν and S k,ν , for k = 1 , are realfunctions depending smoothly on x , y and y and that R ,ν and S ,ν are given by:(5.2) R ,ν = − α ν x + α ν y + O (cid:0) | z | + y + | y |k z k (cid:1) ,S ,ν = − α ν x y + O (cid:0) | z | + y + | y |k z k (cid:1) . We write:(5.3) R ,ν = r ,ν x + r ,ν x y + r ,ν y + r ,ν x + r ,ν x y + r ,ν x y + r ,ν y + O (cid:0) | z | + y + | y |k z k (cid:1) S ,ν = s ,ν x + s ,ν x y + s ,ν y + s ,ν x + s ,ν x y + s ,ν x y + s ,ν y + O (cid:0) | z | + y + | y |k z k (cid:1) . It follows that: ρ ν ◦ Ψ − ν ( z , z ) = ℜ ez + β ν | z | + m X k =3 P ′ k ( z , z , ν ) + O (cid:0) | z | m +1 + | z |k z k (cid:1) . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 31
Then we define τ ν := min (cid:18) δ ν | β ν | (cid:19) , min k =3 , ··· , m − (cid:18) δ ν k P ′ k ( ., ν ) k (cid:19) k , δ m ν ! . And we consider the following anisotropic dilations of C : Λ ν ( z , z ) := (cid:0) τ − ν z , δ − ν z (cid:1) . If we write J ν := (Ψ ν ) ∗ J ν as: J ν = (cid:18) J ,ν B ,ν C ,ν J ,ν (cid:19) with C ,ν := (cid:18) ( J ν ) ( J ν ) ( J ν ) ( J ν ) (cid:19) , then we have: (Λ ν ) ∗ J ν ( z ) = (cid:18) J ,ν ( τ ν z , δ ν z ) τ − ν δ ν B ,ν ( τ ν z , δ ν z ) τ ν δ − ν C ,ν ( τ ν z , δ ν z ) J ,ν ( τ ν z , δ ν z ) (cid:19) . We have generically the following situation:
Proposition 5.1.
The sequence of structures (Λ ν ) ∗ J ν converges to the standard structure J st if and only ifthe D’Angelo type of the origin is less than or equal to four.Proof. We notice that (Λ ν ) ∗ J ν converges to J st if and only if C ,ν = O (cid:0) | z | m − (cid:1) + O ( | z | ) . Indeed if C ,ν = O (cid:0) | z | m − (cid:1) + O ( | z | ) then τ ν δ − ν C ,ν ( τ ν z , δ ν z ) = τ mν δ − ν O | z | m + τ mν O | z | m , which converges to the zero by matrix since τ ν ≤ δ m ν and since C ,ν tends to the zero by matrix.Conversely if C ,ν = O (cid:0) | z | k (cid:1) + O ( | z | ) , with k < m − , then (Λ ν ) ∗ J ν converges to a polynomialintegrable structure ˜ J = J st + O | z | wich is generically different from J st .We have proved in Lemma 4.2 that when the origin is a point of D’Angelo type four, then C ,ν = O (cid:0) | z | (cid:1) + O ( | z | ) and so (Λ ν ) ∗ J ν = (Λ ν ◦ Ψ ν ) ∗ J ν converges to J st when ν tends to + ∞ , with: R ,ν = S ,ν = 0 ,R ,ν = − α ν x + α ν y ,S ,ν = − α ν x y . In case the D’Angelo type of the origin is greater than four, we cannot guarantee the convergence of τ ν δ − ν C ν ( τ ν z , δ ν z ) when we only remove harmonic terms. So we need to find a more general sequenceof diffeomorphisms Ψ ν defined by (5.1), (5.2) and (5.3) and such that C ,ν = O (cid:0) | z | m − (cid:1) + O ( | z | ) . Claim.
There are no polynomial R ,ν , S ,ν , R ,ν and S ,ν such that C ,ν does not contain any order threeterms in x and y . A direct computation leads to: α − ν ( J ν ) ( z ) = ( a ν − a ν ) (cid:0) Ψ − ν ( z ) (cid:1) x − ( c ν + b ν ) (cid:0) Ψ − ν ( z ) (cid:1) y − y ∂R ,ν ∂x − x ∂R ,ν ∂y − x ∂S ,ν ∂x + y ∂S ,ν ∂y + x ∂R ,ν ∂x ∂S ,ν ∂x + y ∂R ,ν ∂x ∂S ,ν ∂y − y ∂R ,ν ∂y ∂S ,ν ∂x + x ∂R ,ν ∂y ∂S ,ν ∂y − y (cid:18) ∂S ,ν ∂x (cid:19) − y (cid:18) ∂S ,ν ∂y (cid:19) − x ∂R ,ν ∂x ∂R ,ν ∂y + x ∂S ,ν ∂y ∂R ,ν ∂y + y ∂R ,ν ∂y ∂R ,ν ∂y + y ∂S ,ν ∂x ∂R ,ν ∂y + O (cid:0) | z | + | z |k z k (cid:1) and to α − ν ( J ν ) ( z ) = ( b ν − b ν ) (cid:0) Ψ − ν ( z ) (cid:1) x + ( a ν + a ν ) (cid:0) Ψ − ν ( z ) (cid:1) y + x ∂R ,ν ∂x − y ∂R ,ν ∂y − y ∂S ,ν ∂x − x ∂S ,ν ∂y − x (cid:18) ∂R ,ν ∂x (cid:19) − x (cid:18) ∂R ,ν ∂y (cid:19) + y ∂R ,ν ∂x ∂S ,ν ∂x + x ∂R ,ν ∂x ∂S ,ν ∂y − x ∂R ,ν ∂y ∂S ,ν ∂x + y ∂R ,ν ∂y ∂S ,ν ∂y − x ∂R ,ν ∂y ∂R ,ν ∂y − x ∂S ,ν ∂x ∂R ,ν ∂y − y ∂R ,ν ∂x ∂R ,ν ∂y + y ∂S ,ν ∂y ∂R ,ν ∂y + O (cid:0) | z | + | z |k z k (cid:1) . The only order two terms in x and y of α − ν ( J ν ) ( z ) and of α − ν ( J ν ) ( z ) are those contained, respec-tively, in − y ∂R ,ν ∂x − x ∂R ,ν ∂y − x ∂S ,ν ∂x + y ∂S ,ν ∂y and x ∂R ,ν ∂x − y ∂R ,ν ∂y − y ∂S ,ν ∂x − x ∂S ,ν ∂y . Vanishing these order two terms leads to: R ,ν = r ,ν x − s ,ν x y − r ,ν y + r ,ν x + r ,ν x y + r ,ν x y + r ,ν y + O (cid:0) | z | + y + | y |k z k (cid:1) S ,ν = s ,ν x + 2 s ,ν x y − s ,ν y + s ,ν x + s ,ν x y + s ,ν x y + s ,ν y + O (cid:0) | z | + y + | y |k z k (cid:1) . SEUDOCONVEX REGIONS OF FINITE D’ANGELO TYPE 33
Then it follows that: α − ν ( J ν ) ( z ) = ( a ν − a ν ) (cid:0) Ψ − ν ( z ) (cid:1) x − ( c ν + b ν ) (cid:0) Ψ − ν ( z ) (cid:1) y − y ∂R ,ν ∂x − x ∂R ,ν ∂y − x ∂S ,ν ∂x + y ∂S ,ν ∂y + O (cid:0) | z | + | z |k z k (cid:1) , and that α − ν ( J ν ) ( z ) = ( b ν − b ν ) (cid:0) Ψ − ν ( z ) (cid:1) x + ( a ν + a ν ) (cid:0) Ψ − ν ( z ) (cid:1) y + x ∂R ,ν ∂x − y ∂R ,ν ∂y − y ∂S ,ν ∂x − x ∂S ,ν ∂y + O (cid:0) | z | + | z |k z k (cid:1) . Since J ν satisfies (2.3), we have: ( a ν − a ν ) (cid:0) Ψ − ν ( z ) (cid:1) x − ( c ν + b ν ) (cid:0) Ψ − ν ( z ) (cid:1) y = H ,ν ( x , y ) + O (cid:0) | z | + | z |k z k (cid:1) ( b ν − b ν ) (cid:0) Ψ − ν ( z ) (cid:1) x + ( a ν + a ν ) (cid:0) Ψ − ν ( z ) (cid:1) y = H ′ ,ν ( x , y ) + O (cid:0) | z | + | z |k z k (cid:1) , where H ,ν ( x , y ) and H ′ ,ν ( x , y ) are real homogeneous polynomials of degree three in x and y whichare generically non identically zero. Since we cannot insure the convergence of α ν τ ν δ − ν H ,ν ( τ ν x , τ ν y ) = α ν τ ν δ − ν H ,ν ( x , y ) and α ν τ ν δ − ν H ′ ,ν ( τ ν x , τ ν y ) = α ν τ ν δ − ν H ′ ,ν ( x , y ) , we want to cancel polynomials H ,ν ( x , y ) and H ′ ,ν ( x , y ) by order three terms in x and y containedin − y ∂R ,ν ∂x − x ∂R ,ν ∂y − x ∂S ,ν ∂x + y ∂S ,ν ∂y and x ∂R ,ν ∂x − y ∂R ,ν ∂y − y ∂S ,ν ∂x − x ∂S ,ν ∂y . Finally, vanishing order three terms in x and y of α − ν ( J ν ) ( z ) and of α − ν ( J ν ) ( z ) involve thefollowing system of linear equations: − − − − −
30 0 1 0 0 2 0 30 0 0 3 0 0 1 0 r ,ν r ,ν r ,ν r ,ν s ,ν s ,ν s ,ν s ,ν = Y Since this × system of linear equations is not a Cramer system, it follows that there does not exist,generically, polynomials R ,ν and S ,ν such that there are no order three term in x and y in ( J ν ) ( z ) and ( J ν ) ( z ) . (cid:3) R EFERENCES [1] J.-F.Barraud, E.Mazzilli,
Regular type of real hyper-surfaces in (almost) complex manifolds , Math. Z., 248 (2004), no. 4,757-772.[2] F.Berteloot,
Attraction des disques analytiques et continuit´e hold´erienne d’applications holomorphes propres , Topics in com-plex analysis (Warsaw, 1992), Banach Center Publ., 31, Polish Acad. Sci., Warsaw, 1995, 91-98.[3] F.Berteloot,
Principe de Bloch et estimations de la m´etrique de Kobayashi dans les domaines de C , J. Geom. Anal., 13-1(2003), 29-37.[4] T.Bloom-I.Graham, A geometric characterization of type on real submanifolds of C n , J. Diff. Geometry, 12 (1977), 171-182.[5] D.Catlin, Estimates of invariant metrics on pseudoconvex domains if dimension two ,Math. Z,200(1989),429-466.[6] B.Coupet, H.Gaussier,A.Sukhov,
Fefferman’s mapping theorem on almost complex manifolds in complex dimension two.
Math. Z., 250 (2005), no. 1, 59-90.[7] J.-P.D’Angelo,
Real hypersurface, orders of contact, and applications
Ann. of Math. 115 (1982), 615-637.[8] J.-P.D’Angelo,
Finite type conditions for real hypersurfaces
J.Diff. Geometry, 14 (1979), 59-66.[9] K.Diederich, A.Sukhov,
Plurisubharmonic exhaustion functions and almost complex Stein structures , preprint, math.CV/.[10] R.Debalme,
Kobayashi hyperbolicity of almost complex manifolds , preprint of the University of Lille, IRMA 50 (1999),math.CV/9805130.[11] J.E.Fornaess, N.Sibony,
Construction of p.s.h. functions on weakly pseudoconvex domains , Duke Math. J., vol. 58 (1989),633-655.[12] H.Gaussier, A.Sukhov,
Estimates of the Kobayashi metric on almost complex manifolds , Bull. Soc. Math. France, 133 (2005),no. 2, 259-273.[13] H.Gaussier, A.Sukhov,
Wong-Rosay Theorem in almost complex manifolds , preprint, math.CV/0412095.[14] I.Graham,
Boundary behaviour of the Caratheodory and Kobayashi metrics on strongly pseudoconvex domains in C n withsmooth boundary , Trans. Amer. Math. Soc. 207 (1975), 219-240.[15] M.Gromov, Pseudoholomorphic curves in symplectic manifolds , Invent. Math. 82-2 (1985), 307-347.[16] F.Haggui,
Fonctions PSH sur une vari´et´e presque complexe , C. R. Acad. Sci. Paris, Ser.I 335 (2002), 1-6.[17] S.Ivashkovich, J.-P.Rosay
Schwarz-type lemmas for solutions of ∂ -inequalities and complete hyperbolicity of almost complexmanifolds , Ann. Inst. Fourier 54 (2004), no. 7, 2387-2435.[18] J.Kohn, Boundary behavior of ∂ on weakly pseudoconvex manifolds of dimension two , J.Diff Geometry, 6 (1972), 523-542.[19] B.Kruglikov, Existence of close pseudoholomorphic disks for almost complex manifolds and their application to theKobayashi-Royden pseudonorm , (Russian) Funktsional. Anal. i Prilozhen. 33 (1999), no. 1, 46-58, 96; translation in Funct.Anal. Appl. 33 (1999), no. 1, 38-48.[20] K.H.Lee,
Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex bound-ary point. , Michigan Math. J., 54 (2006), no. 1, 179-205.[21] A.Nijenhuis, W.Woolf,
Some integration problems in almost-complex and complex manifolds , Ann. Math. 77(1963), 429-484.[22] N.Sibony,
A class of hyperbolic manifolds ,Ann. of Math. Stud., 100, Princeton Univ. Press, Princeton, NJ, 1981., 91-97[23] J.-C.Sikorav,
Some properties of holomorphic curves in almost complex manifolds , Holomorphic Curves in Symplectic Ge-ometry, eds. M. Audin and J. Lafontaine, Birkhauser (1994), 165-189.LATP, C.M.I, 39
RUE J OLIOT -C URIE
ARSEILLE CEDEX
13, FRANCE
E-mail address ::